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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Euler's four-square identity ： ウィキペディア英語版 Euler's four-square identity In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. Specifically: ::$(a\_1^2+a\_2^2+a\_3^2+a\_4^2)(b\_1^2+b\_2^2+b\_3^2+b\_4^2)=\backslash ,$ ::$(a\_1\; b\_1\; +\; a\_2\; b\_2\; +\; a\_3\; b\_3\; +\; a\_4\; b\_4)^2\; +\backslash ,$ ::$(a\_1\; b\_2\; -\; a\_2\; b\_1\; +\; a\_3\; b\_4\; -\; a\_4\; b\_3)^2\; +\backslash ,$ ::$(a\_1\; b\_3\; -\; a\_2\; b\_4\; -\; a\_3\; b\_1\; +\; a\_4\; b\_2)^2\; +\backslash ,$ ::$(a\_1\; b\_4\; +\; a\_2\; b\_3\; -\; a\_3\; b\_2\; -\; a\_4\; b\_1)^2.\backslash ,$ Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach〔''Leonhard Euler: Life, Work and Legacy'', R.E. Bradley and C.E. Sandifer (eds), Elsevier, 2007, p. 193〕〔''Mathematical Evolutions'', A. Shenitzer and J. Stillwell (eds), Math. Assoc. America, 2002, p. 174〕 (but he used a different sign convention from the above). It can be proven with elementary algebra and holds in every commutative ring. If the $a\_k$ and $b\_k$ are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers. The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any $a\_k$ to $-a\_k$, $b\_k$ to $-b\_k$, or by changing the signs inside any of the squared terms on the right hand side. Hurwitz's theorem states that an identity of form, :$(a\_1^2+a\_2^2+a\_3^2+...+a\_n^2)(b\_1^2+b\_2^2+b\_3^2+...+b\_n^2)\; =\; c\_1^2+c\_2^2+c\_3^2+...+c\_n^2\backslash ,$ where the $c\_i$ are bilinear functions of the $a\_i$ and $b\_i$ is possible only for ''n'' = . However, the more general Pfister's theorem allows that if the $c\_i$ are just rational functions of one set of variables, hence has a denominator, then it is possible for all $n\; =\; 2^m$.〔Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf〕 Thus, a different kind of four-square identity can be given as, :$(a\_1^2+a\_2^2+a\_3^2+a\_4^2)(b\_1^2+b\_2^2+b\_3^2+b\_4^2)=\backslash ,$ ::$(a\_1\; b\_4\; +\; a\_2\; b\_3\; +\; a\_3\; b\_2\; +\; a\_4\; b\_1)^2\; +\backslash ,$ ::$(a\_1\; b\_3\; -\; a\_2\; b\_4\; +\; a\_3\; b\_1\; -\; a\_4\; b\_2)^2\; +\backslash ,$ ::$\backslash left(a\_1\; b\_2\; +\; a\_2\; b\_1\; +\; \backslash frac\; -\; \backslash frac\backslash right)^2+\backslash ,$ ::$\backslash left(a\_1\; b\_1\; -\; a\_2\; b\_2\; -\; \backslash frac\; -\; \backslash frac\backslash right)^2\backslash ,$ where, :$u\_1\; =\; b\_1^2b\_4-2b\_1b\_2b\_3-b\_2^2b\_4$ :$u\_2\; =\; b\_1^2b\_3+2b\_1b\_2b\_4-b\_2^2b\_3$ Note also the incidental fact that, :$u\_1^2+u\_2^2\; =\; (b\_1^2+b\_2^2)^2(b\_3^2+b\_4^2)$ ==See also== *Brahmagupta–Fibonacci identity (sums of two squares) *Degen's eight-square identity *Pfister's sixteen-square identity *Latin square 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euler's four-square identity」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.